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%I #15 Jul 22 2020 13:26:01
%S 1,1,3,1,4,3,4,1,6,4,5,3,5,4,7,1,7,6,8,4,7,5,6,3,9,5,10,4,9,7,8,1,8,7,
%T 9,6,9,8,8,4,9,7,10,5,11,6,7,3,9,9,11,5,13,10,10,4,12,9,10,7,9,8,11,1,
%U 10,8,10,7,9,9,10,6,10,9,13,8,11,8,10,4,15,9,10,7,13,10,13,5,14,11,10,6,12,7,15,3,10,9,12,9,15,11,14,5,13
%N Number of distinct integers encountered on all possible paths from n to any first encountered powers of 2 (that are included in the count), when using the transitions x -> x - (x/p) and x -> x + (x/p) in any order, where p is the largest prime dividing x.
%H Antti Karttunen, <a href="/A335906/b335906.txt">Table of n, a(n) for n = 1..65537</a>
%e From 9 one can reach with the transitions x -> A171462(x) (leftward arrow) and x -> A335876(x) (rightward arrow) the following six numbers, when one doesn't expand any power of 2 further:
%e 9
%e / \
%e 6 12
%e / \ / \
%e 4 8 16
%e thus a(9) = 6.
%e From 10 one can reach with the transitions x -> A171462(x) and x -> A335876(x) the following for numbers, when one doesn't expand any power of 2 further:
%e 10
%e |\
%e | \
%e | 12
%e | /\
%e |/ \
%e 8 16
%e thus a(10) = 4.
%e From 15 one can reach with the transitions x -> A171462(x) and x -> A335876(x) the following seven numbers, when one doesn't expand any power of 2 further:
%e 15
%e / \
%e / \
%e 12<----18
%e / \ \
%e / \ \
%e 8 16<----24
%e \
%e \
%e 32
%e thus a(15) = 7.
%o (PARI)
%o A171462(n) = if(1==n,0,(n-(n/vecmax(factor(n)[, 1]))));
%o A335876(n) = if(1==n,2,(n+(n/vecmax(factor(n)[, 1]))));
%o A209229(n) = (n && !bitand(n,n-1));
%o A335906(n) = { my(xs=Set([n]),allxs=xs,newxs,a,b,u); for(k=1,oo, newxs=Set([]); if(!#xs, return(#allxs)); allxs = setunion(allxs,xs); for(i=1,#xs,u = xs[i]; if(!A209229(u), newxs = setunion([A171462(u)],newxs); newxs = setunion([A335876(u)],newxs))); xs = newxs); };
%Y Cf. A006530, A171462, A335876, A335905.
%K nonn
%O 1,3
%A _Antti Karttunen_, Jun 30 2020
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